Scarred patterns in surface waves

Arshad Kudrolli, Mathew Abrahams, and Jerry P. Gollub

Phys. Rev. E 63, 026208 (2001). also:

High resolution postscript figures: Fig.1. Fig.2. Fig.3. Fig.4. Fig.5. Fig.6. Fig.7. Fig.8. Fig.9. Fig.10. preprint with figures.

* Abstract: Surface wave patterns are investigated experimentally in a system geometry that has become a paradigm of quantum chaos: the stadium billiard. Linear waves in bounded geometries for which classical ray trajectories are chaotic are known to give rise to scarred patterns. Here, we utilize parametrically forced surface waves (Faraday waves), which become progressively nonlinear beyond the wave instability threshold, to investigate the subtle interplay between boundaries and nonlinearity. Only a subset (three main types) of the computed linear modes of the stadium are observed in a systematic scan. These correspond to modes in which the wave amplitudes are strongly enhanced along paths corresponding to certain periodic ray orbits. The suppression of certain other modes has been shown by Agam and Altshuler in a related theoretical study to be a consequence of higher boundary dissipation relative to the modes that are observed. Spatially asymmetric or disordered (but time-independent) patterns are also found even near onset. As the driving acceleration is increased, the time-independent scarred patterns persist, but in some cases transitions between modes are noted. The onset of spatiotemporal chaos at higher forcing amplitude often involves a nonperiodic oscillation between spatially ordered and disordered states. We have characterized this phenomenon using the concept of pattern entropy. The rate of change of the patterns is found to be reduced as the state passes temporarily near the ordered configurations of lower entropy. We also have found complex but highly symmetric (time-independent) patterns far above onset in the regime that is normally chaotic.

Movies of time-dependent surface wave patterns

Aperiodic oscillation between spatially ordered and disordered patterns is observed (driving frequency = 55 Hz).

Interesting Quantum Chaos related links:

Simulation of ray trajectories in a stadium (and circle.) Link.

Please send comments to:
Arshad Kudrolli

and Jerry Gollub

Last modified Feb. 24, 2000.